Correlation of sum and difference with equal variances

Question

$X$ and $Y$ are random variables (not necessarily independent) with finite means and both with variance $16$ . Find the correlation between $X+Y$ and $X-Y$ .

Short answer

$0$ . Expanding bilinearly, $\operatorname{Cov}(X+Y,X-Y)=\operatorname{Var}X-\operatorname{Var}Y=0$ , since the cross terms cancel. Equal variances make the sum and difference uncorrelated (assuming neither is degenerate).

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Lessons behind this question

Covariance and Correlation
Populations, Samples, and Sampling Distributions

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